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Xuexiu Zhong

Publications and source records attributed to Xuexiu Zhong.

At least 19 recordsLinked to original sources
arXiv2026

Liouville theorems and symmetry of positive solutions for partially confined nonlinear Schrödinger equations

We study positive solutions of the partially confined stationary nonlinear Schrödinger equation $$-Δu+|y|^2u+λu=g(u),\quad (y,z)\in\mathbb{R}^d\times\mathbb{R}^{m},\quad 1\leq d -d$, we establish the existence of positive solutions under some standard assumptions. Furthermore, every positive solution decaying at infinity is radially symmetric and strictly decreasing in the confined variables and, up to one common translation, radially symmetric and strictly decreasing in the free variables. \vskip 0.2in Dedicated to our supervisor Prof. Wenming Zou on the occasion of his 60th birthday.

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arXiv2026

Serrin-type Overdetermination in Scaling Limits: Sharp Existence and Asymptotic Behavior of Quasi-linear Schrödinger Energy Ground States

This paper establishes optimal existence results and limiting profiles for energy ground states of the quasi-linear Schrödinger equation $$ -Δu - Δ(|u|^{2})u + λu = |u|^{p-2}u \quad \text{in } \mathbb{R}^N $$ with prescribed mass $\int_{\mathbb{R}^N}|u|^2 = a > 0$, in the mass-supercritical case $4 + \frac{4}{N} < p < 2 \cdot 2^*$. Breakthrough in existence theory: For all dimensions $N \geq 1$, we completely resolve the existence problem: For $1 \leq N \leq 4$, ground states exist for all $a > 0$. For $N \geq 5$, there exists a sharp threshold $a_0 > 0$ such that ground states exist if and only if $a \leq a_0$. This constitutes the optimal existence theory, crucially removing the restrictive condition $p \leq 2^*$ required in all prior works (which limited results to $N \leq 3$). Asymptotic behavior and new phenomena: We provide a complete asymptotic analysis of normalized ground states: As $a \to 0^+$, solutions exhibit a novel connection to Serrin-type overdetermined problems. Through a delicate rescaling, profiles converge to the unique positive radial solution of the overdetermined problem (the first such result for quasi-linear equations). As $a \to a^*$ ($a^* = \infty$ for $N \leq 4$; $a^* = a_0$ for $N \geq 5$), solutions converge to distinct limiting profiles depending on dimension and nonlinearity. Our methods introduce a new constraint approach and unified variational framework for quasi-linear problems with $L^2$-constraints.

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arXiv2026

Multiple positive solutions with prescribed masses for a coupled Schrödinger system: mass mixed and Sobolev critical coupled case

The aim of this paper is to establish multiple positive normalized solutions $(u,v,λ_1,λ_2)\in H^1(\mathbb{R}^N,\mathbb{R}^2)\times \mathbb{R}^2$ to the following coupled Schrödinger system involving Sobolev critical exponent: $$ \begin{cases} -Δu+λ_1 u=μ_1|u|^{p-2}u+να|u|^{α-2}u|v|^β, x\in \mathbb{R}^N,\\ -Δv+λ_2 v=μ_2|v|^{q-2}v+νβ|v|^{β-2}v|u|^α, x\in \mathbb{R}^N,\\ \int_{\mathbb{R}^N}|u|^2\mathrm{d}x=a, \int_{\mathbb{R}^N}|v|^2\mathrm{d}x=b, \end{cases} N\geq 3, $$ where $μ_1,μ_2, ν, a, b>0$. We are particularly interested in the mass mixed case that $2 1, β>1$, and $α+β=2^*:=\frac{2N}{N-2}$. For sufficiently small $ν>0$, we demonstrate that the above system admits two positive solutions, one of which serves as a local minimizer, and the other as a mountain pass solution. By developing some new technical lemmas on the interaction estimates, we are managed to resolves Soave's open problem [{\it J. Funct. Anal.}, 2020, Remark 1.1] within the context of the system case. Notably, our existence result holds true for all dimensions $N\geq 3$. Our results also significantly extend the result of Gou and Jeanjean [{\it Nonlinearity}, 2018, Theorem 1.1] to the Sobolev critical coupled case and removing the hypothesis ``either $p,q\leq α+β-\frac{2}{N}$ or $|p-q|\leq \frac{2}{N}$" for $N\geq 5$. Additionally, we also establish a sequence of properties for the local minimizer, including local uniqueness, continuity with respect to the small parameter $ν$, and the limiting profiles for $ν\rightarrow 0^+$.

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arXiv2026

Sharp interaction estimates and their application: existence of normalized ground states to coupled Schrödinger systems with potentials

In this paper, our aim is to prove the existence of normalized ground state for the following Schrödinger systems with potentials $$\begin{cases} -Δu_1+V_1(x)u_1+λ_1 u_1=\partial_1 G(u_1,u_2)\;\quad&\hbox{in}\;\mathbb{R}^N,\\ -Δu_2+V_2(x)u_2+λ_2 u_2=\partial_2G(u_1,u_2)\;\quad&\hbox{in}\;\mathbb{R}^N,\\ 0 -\infty$, which are allowed to be singular at some points. And the nonlinearities $G(u_1,u_2)$ are considered of the form $$ \begin{cases} G(u_1, u_2):=\sum_{i=1}^{\ell}\frac{μ_i}{p_i}|u_1|^{p_i}+\sum_{j=1}^{m}\frac{ν_j}{q_j}|u_2|^{q_j}+\sum_{k=1}^{n}β_k |u_1|^{r_{1,k}}|u_2|^{r_{2,k}},~~\ell,m,n\in \mathbb{N}^+_0, μ_i, ν_j,β_k>0, ~2 1, i=1,2,\cdots, \ell; j=1,2,\cdots, m; k=1,2,\cdots, n. \end{cases} $$ Under the mass sub-critical assumption, the normalized ground states are obtained as the minimum of the functional $J$ on the manifold $S_{a_1,a_2}$. Since the functional is not weak lower semi-continuous, to prove the minimizing problem is achievable, the key step is establishing the strict sub-additive inequality. Among its main ingredients is the study of the sharp decay of the positive solutions and the interaction estimates.

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arXiv2026

Bivariate Hardy-Sobolev Inequality and Its Sharp Stability

This paper establishes a bivariate Hardy-Sobolev inequality. Let $Ω\subset \mathbb{R}^N$ ($N \geq 3$) be an open domain, $s \in (0,2)$, $α> 1$, $β> 1$ with $α+ β= 2^*(s)$, and $κ\in \mathbb{R}$. For any functions $u, v \in D_0^{1,2}(Ω)$, we prove the inequality: \begin{multline*} \int_Ω |\nabla u|^2 \, \mathrm{d}x + \int_Ω |\nabla v|^2 \, \mathrm{d}x \ge S_{α,β,λ,μ}(Ω) \left( \int_Ω \Big( λ\frac{|u|^{2^*(s)}}{|x|^s} + μ\frac{|v|^{2^*(s)}}{|x|^s} + 2^*(s) κ\frac{|u|^α|v|^β}{|x|^s} \Big)\, \mathrm{d}x \right)^{\frac{2}{2^*(s)}}. \end{multline*} We derive the best constant $S_{α,β,λ,μ}(Ω)$ and characterize the set of minimizers. Moreover, for $Ω= \mathbb{R}^N$ and $κ> 0$, we obtain sharp stability results for nonnegative functions.

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arXiv2026

Uniqueness of radial solutions for $m$-Laplacian equations in low dimensions

This paper extends the uniqueness results of Serrin and Tang [\textit{Indiana Univ. Math. J.}, 49 (2000), pp. 897--923] to the low-dimensional case $1\leq N\leq m$ with $m>1$. We consider radial solutions of the overdetermined problem \[ \begin{cases} -Δ_m u = f(u), \quad u>0 & \text{in } B_R,\\[4pt] u = \partial_νu = 0 & \text{on } \partial B_R, \text{ if } R<\infty,\\[4pt] \displaystyle\lim_{|x|\to\infty} u(x)=0, & \text{if } R=\infty, \end{cases} \] where $B_R$ is the open ball in $\mathbb{R}^N$ centered at the origin with radius $R>0$ (the case $R=\infty$ corresponds to the whole space, for studying positive ground states). Under suitable assumptions on the nonlinearity $f$, we establish the uniqueness of such solutions, whenever they exist. Our analysis is motivated by connections to sharp forms of the Gagliardo--Nirenberg and Nash inequalities. Although the overall framework follows that of Serrin and Tang, the details of our proofs differ substantially in the low-dimensional setting. In particular, Serrin and Tang explicitly noted that their techniques rely heavily on the condition $N>m$ and do not readily extend to $N\leq m$ (see Subsection~6.2 of their work). The present paper closes this gap, thereby providing a complete uniqueness theory for all dimensions. As a concrete example, for the canonical nonlinearity $f(u) = -u^p + u^q$ with $p m$ and $m^* = \infty$ for $N\leq m$. Consequently, our work also completely resolves an open problem posed by Pucci and Serrin [\textit{Indiana Univ. Math. J.}, 47 (1998), pp. 501--528], which had been settled for $N>m$ in the earlier work of Serrin and Tang.

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arXiv2024

Normalized solutions to Kirchhoff type equations with a critical growth nonlinearity

In this paper, we are concerned with normalized solutions of the Kirchhoff type equation \begin{equation*} -M\left(\int_{\R^N}|\nabla u|^2\mathrm{d} x\right)Δu = λu +f(u) \ \ \mathrm{in} \ \ \mathbb{R}^N \end{equation*} with $u \in S_c:=\left\{u \in H^1(\R^N): \int_{\R^N}u^2 \mathrm{d}x=c^2\right\}$. When $N=2$ and $f$ has exponential critical growth at infinity, normalized mountain pass type solutions are obtained via the variational methods. When $N \ge 4$, $M(t)=a+bt$ with $a$, $b>0$ and $f$ has Sobolev critical growth at infinity, we investigate the existence of normalized ground state solutions and normalized mountain pass type solutions. Moreover, the non-existence of normalized solutions is also considered.

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arXiv2024

The mass-mixed case for normalized solutions to NLS equations in dimension two

\noindent We are concerned with positive normalized solutions $(u,λ)\in H^1(\mathbb{R}^2)\times\mathbb{R}$ to the following semi-linear Schrödinger equations $$ -Δu+λu=f(u), \quad\text{in}~\mathbb{R}^2, $$ satisfying the mass constraint $$\int_{\mathbb{R}^2}|u|^2\, dx=c^2\ .$$ We are interested in the so-called mass mixed case in which $f$ has $L^2$-subcritical growth at zero and critical growth at infinity, which in dimension two turns out to be of exponential rate. Under mild conditions, we establish the existence of two positive normalized solutions provided the prescribed mass is sufficiently small: one is a local minimizer and the second one is of mountain pass type. We also investigate the asymptotic behavior of solutions approaching the zero mass case, namely when $c\to 0^+$.

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arXiv2024

Single-peak and multi-peak solutions for Hamiltonian elliptic systems in dimension two

This paper is concerned with the Hamiltonian elliptic system in dimension two\begin{equation*}\aligned \left\{ \begin{array}{lll} -ε^2Δu+V(x)u=g(v)\ & \text{in}\quad \mathbb{R}^2,\\ -ε^2Δv+V(x)v=f(u)\ & \text{in}\quad \mathbb{R}^2, \end{array}\right.\endaligned \end{equation*} where $V\in C(\mathbb{R}^2)$ has local minimum points, and $f,g\in C^1(\mathbb{R})$ are assumed to be of exponential growth in the sense of Trudinger-Moser inequality. When $V$ admits one or several local strict minimum points, we show the existence and concentration of single-peak and multi-peak semiclassical states respectively, as well as strong convergence and exponential decay. In addition, positivity of solutions and uniqueness of local maximum points of solutions are also studied. Our theorems extend the results of Ramos and Tavares [Calc. Var. 31 (2008) 1-25], where $f$ and $g$ have polynomial growth. It seems that it is the first attempt to obtain multi-peak semiclassical states for Hamiltonian elliptic system with exponential growth.

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arXiv2024

Normalized solutions of quasilinear Schrödinger equations with a general nonlinearity

We are concerned with solutions of the following quasilinear Schrödinger equations \begin{eqnarray*} -{\mathrm{div}}\left(φ^{2}(u) \nabla u\right)+φ(u) φ^{\prime}(u)|\nabla u|^{2}+λu=f(u), \quad x \in \mathbb{R}^{N} \end{eqnarray*} with prescribed mass $$ \int_{\mathbb{R}^{N}} u^{2} \mathrm{d}x=c, $$ where $N\ge 3, c>0$, $λ\in \mathbb{R}$ appears as the Lagrange multiplier and $φ\in C ^{1}(\mathbb{R} ,\mathbb{R}^{+})$. The nonlinearity $f \in C\left ( \mathbb{R}, \, \mathbb{R} \right )$ is allowed to be mass-subcritical, mass-critical and mass-supercritical at origin and infinity. Via a dual approach, the fixed point index and a global branch approach, we establish the existence of normalized solutions to the problem above. The results extend previous results by L. Jeanjean, J. J. Zhang and X.X. Zhong to the quasilinear case.

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arXiv2023

On the Kirchhoff equation with prescribed mass and general nonlinearities

In the present paper, we apply a global branch approach to study the existence, non-existence and multiplicity of positive normalized solutions $(λ_c, u_c)\in \mathbb{R}\times H^1(\mathbb{R}^N)$ to the following Kirchhoff problem $$ -\left(a+b\int_{\mathbb{R}^N}|\nabla u|^2dx\right)Δu+λu=g(u)~\hbox{in}~\mathbb{R}^N,\;N\geq 1 $$ satisfying the normalization constraint $ \displaystyle\int_{\mathbb{R}^N}u^2=c, $ which appears in free vibrations of elastic strings. The parameters $a,b>0$ are prescribed as well as the mass $c>0$. Due to the presence of the non-local term $b\int_{\mathbb{R}^N}|\nabla u|^2dx Δu$, such problems lack the mountain pass geometry in the higher dimension case $N\geq 5$. Our result seems to be the first attempt in this aspect.

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arXiv2023

Normalized ground states for a coupled Schrödinger system: Mass super-critical case

We consider the existence of solutions $(λ_1,λ_2, u, v)\in \mathbb{R}^2\times (H^1(\mathbb{R}^N))^2$ to systems of coupled Schrödinger equations $$ \begin{cases} -Δu+λ_1 u=μ_1 u^{p-1}+βr_1 u^{r_1-1}v^{r_2}\quad &\hbox{in}~\mathbb{R}^N,\\ -Δv+λ_2 v=μ_2 v^{q-1}+βr_2 u^{r_1}v^{r_2-1}\quad &\hbox{in}~\mathbb{R}^N,\\ 0 0$ and the prescribed masses $a,b>0$. We focus on the coupled purely mass super-critical case, i.e., $$2+\frac{4}{N} 0$ and $β>0$.

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arXiv2023

A global branch approach to normalized solutions for the Schrödinger equation

We study the existence, non-existence and multiplicity of prescribed mass positive solutions to a Schrödinger equation of the form \begin{equation*} -Δu+λu=g(u), \quad u \in H^1(\mathbb{R}^N), \, N \geq 1. \end{equation*} Our approach permits to handle in a unified way nonlinearities $g(s)$ which are either mass subcritical, mass critical or mass supercritical. Among its main ingredients is the study of the asymptotic behaviors of the positive solutions as $λ\rightarrow 0^+$ or $λ\rightarrow +\infty$ and the existence of an unbounded continuum of solutions in $(0, + \infty) \times H^1(\mathbb{R}^N)$.

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arXiv2023

Normalized solutions for critical Choquard systems

In this paper, we consider the critical Choquard system with prescribed mass \begin{equation*} \begin{aligned} \left\{ \begin{array}{lll} -Δu+λ_1u=(I_μ\ast |u|^{2^*_μ})|u|^{2^*_μ-2}u+νp(I_μ\ast |v|^q)|u|^{p-2}u\ & \text{in}\quad \mathbb{R}^N,\\ -Δv+λ_2v=(I_μ\ast |v|^{2^*_μ})|v|^{2^*_μ-2}v+νq(I_μ\ast |u|^p)|v|^{q-2}v\ & \text{in}\quad \mathbb{R}^N,\\ \int_{\mathbb{R}^N}u^2=a^2,\quad\int_{\mathbb{R}^N}v^2=b^2, \end{array}\right.\end{aligned} \end{equation*} where $N\geq3$, $0<μ 0$, we study the existence, non-existence and asymptotic behavior of normalized solutions by distinguishing three cases: $L^2$-subcritical case: $p+q<4+\frac{4-2μ}{N}$; $L^2$-critical case: $p+q=4+\frac{4-2μ}{N}$; $L^2$-supercritical case: $p+q>4+\frac{4-2μ}{N}$. In particular, in $L^2$-subcritical case, and either $N\in\{3,4\}$ or $N\geq5$ with $(\frac N2-1)p+\frac {N}{2}q\leq 2N-μ$ and $(\frac N2-1)q+\frac {N}{2}p\leq 2N-μ$, we prove that there exists $ν_0>0$ such that the system has a positive radial normalized ground state for $0<ν<ν_0$. In $L^2$-critical case and $N\in\{3,4\}$, we show there is $ν'_0>0$ such that the system has a positive radial normalized ground state for $0<ν<ν'_0$. In $L^2$-supercritical case and $N\in\{3,4\}$, there are two thresholds $ν_2\geqν_1\geq0$ such that a positive radial normalized solution exists if $ν>ν_2$, and no normalized ground state exists for $ν<ν_1$.

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arXiv2023

Normalized solutions for a Kirchhoff type equations with potential in $\mathbb{R}^3$

In the present paper, we study the existence of normalized solutions to the following Kirchhoff type equations \begin{equation*} -\left(a+b\int_{\R^3}|\nabla u|^2\right)Δu+V(x)u+λu=g(u)~\hbox{in}~\R^3 \end{equation*} satisfying the normalized constraint $\displaystyle\int_{\R^3}u^2=c$, where $a,b,c>0$ are prescribed constants, and the nonlinearities $g(u)$ are very general and of mass super-critical. Under some suitable assumptions on $V(x)$ and $g(u)$, we can prove the existence of ground state normalized solutions $(u_c, λ_c)\in H^1(\R^3)\times\mathbb{R}$, for any given $c>0$. Due to the presence of the nonlocal term, the weak limit $u$ of any $(PS)_C$ sequence $\{w_n\}$ may not belong to the corresponding Pohozaev manifold, which is different from the local problem. So we have to overcome some new difficulties to gain the compactness of a $(PS)_C$ sequence.

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arXiv2022

The existence of positive solution for an elliptic problem with critical growth and logarithmic perturbation

We consider the existence and nonexistence of positive solution for the following Brézis-Nirenberg problem with logarithmic perturbation: \begin{equation*} \begin{cases} -Δu={\left|u\right|}^{{2}^{\ast }-2}u+λu+μu\log {u}^{2} &x\in Ω, \quad \;\:\, u=0& x\in \partial Ω, \end{cases} \end{equation*} where $Ω$ $\subset$ $\R^N$ is a bounded smooth domain, $λ, μ\in \R$, $N\ge3$ and ${2}^{\ast }:=\frac{2N}{N-2}$ is the critical Sobolev exponent for the embedding $H^1_{0}(Ω)\hookrightarrow L^{2^\ast}(Ω)$. The uncertainty of the sign of $s\log s^2$ in $(0, +\infty)$ has some interest in itself. We will show the existence of positive ground state solution which is of mountain pass type provided $λ\in \R, μ>0$ and $N\geq 4$. While the case of $μ<0$ is thornier. However, for $N=3,4$ $λ\in (-\infty, λ_1(Ω))$, we can also establish the existence of positive solution under some further suitable assumptions. And a nonexistence result is also obtained for $μ<0$ and $-\frac{(N-2)μ}{2}+\frac{(N-2)μ}{2}\log(-\frac{(N-2)μ}{2})+λ-λ_1(Ω)\geq 0$ if $N\geq 3$. Comparing with the results in Brézis, H. and Nirenberg, L. (Comm. Pure Appl. Math. 1983), some new interesting phenomenon occurs when the parameter $μ$ on logarithmic perturbation is not zero.

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